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Spectral properties of a certain class of Carleman operators

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*
S. M. Bahri
*

** Address.**
Department of Mathematics, University of
Mostaganem, BP 227,
Mostaganem, 27000, Algeria

** E-mail:**
bahrisidimohamed@univ-mosta.dz

**Abstract.**
The object of the present work is to construct all the generalized spectral
functions of a certain class of Carleman operators in the Hilbert space $
L^{2}( X,\mu ) $ and establish the corresponding expansion
theorems, when the deficiency indices are (1,1). This is done by
constructing the generalized resolvents of $A$ and then using the Stieltjes
inversion formula.

**AMSclassification. ** Primary 05C38, 15A15; Secondary 05A15, 15A18.

**Keywords. ** Spectral theory, integral operator, defect indices.